Determinant formulas

Every formula JEE Advanced expects you to know for Determinant, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 28
Subject: Mathematics
Exam: JEE Advanced
No sign-in needed: ₹0

Determinant, every formula

28 formulas, typeset and free. Print it, or keep it open beside your practice.

Determinant 2×2
$a c b d = a d - b c$
Determinant 3×3 (row expansion)
$∣ A ∣ = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13}$
Minor
$M_{ij} = det of the submatrix deleting row i, col j$
Cofactor
$A_{ij} = (- 1)^{i + j} M_{ij}$
Transpose invariance
$∣ A^{T} ∣ = ∣ A ∣$
Row/column swap
$swap two rows \Rightarrow ∣ A ∣ \to - ∣ A ∣$
Two identical rows
$two equal rows \Rightarrow ∣ A ∣ = 0$
Scalar multiple of a row
$multiply one row by k \Rightarrow ∣ A ∣ \to k ∣ A ∣$
Determinant of kA
$∣ k A ∣ = k^{n} ∣ A ∣$
n = order
Invariant row operation
$R_{i} \to R_{i} + k R_{j} leaves ∣ A ∣ unchanged$
Determinant of a product
$∣ A B ∣ = ∣ A ∣ ∣ B ∣$
Triangular determinant
$triangular \Rightarrow ∣ A ∣ = \prod a_{ii}$
Area of a triangle
$Δ = \frac{1}{2} x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} 111$
Collinearity condition
$x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} 111 = 0$
Cramer's rule
$x = \frac{Δ _{1}}{Δ}, y = \frac{Δ _{2}}{Δ}, z = \frac{Δ _{3}}{Δ}$
Δ≠0
Unique solution
$Δ \neq = 0 \Rightarrow unique solution$
Infinite/consistent
$Δ = 0, Δ_{1} = Δ_{2} = Δ_{3} = 0 \Rightarrow infinitely many (or check)$
Inconsistent
$Δ = 0, some Δ_{i} \neq = 0 \Rightarrow no solution$
Homogeneous non-trivial
$A X = 0 non-trivial \Leftrightarrow Δ = 0$
Determinant of identity
$∣ I_{n} ∣ = 1$
Cross cofactor sum
$a_{i 1} A_{j 1} + a_{i 2} A_{j 2} + a_{i 3} A_{j 3} = 0 (i \neq = j)$
Determinant of adjoint
$∣ adj A ∣ = ∣ A ∣^{n - 1}$
Determinant of inverse
$∣ A^{- 1} ∣ = \frac{1}{∣ A ∣}$
Vandermonde
$111 a b c a^{2} b^{2} c^{2} = (a - b) (b - c) (c - a)$
Skew-symmetric (odd order)
$A^{T} = - A, n odd \Rightarrow ∣ A ∣ = 0$
Differentiation of a determinant
$\frac{d}{d x} ∣ A ∣ = sum of dets differentiating one row at a time$
Row of zeros
$a row of zeros \Rightarrow ∣ A ∣ = 0$
Sum property of a row
$a row = (p + q) \Rightarrow ∣ A ∣ splits into two determinants$

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Determinant formulas

Determinant, every formula

Determinant 2×2

Determinant 3×3 (row expansion)

Minor

Cofactor

Transpose invariance

Row/column swap

Two identical rows

Scalar multiple of a row

Determinant of kA

Invariant row operation

Determinant of a product

Triangular determinant

Area of a triangle

Collinearity condition

Cramer's rule

Unique solution

Infinite/consistent

Inconsistent

Homogeneous non-trivial

Determinant of identity

Cross cofactor sum

Determinant of adjoint

Determinant of inverse

Vandermonde

Skew-symmetric (odd order)

Differentiation of a determinant

Row of zeros

Sum property of a row

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